Minimum clique partition problem with constrained weight for interval graphs

Abstract

Interval graphs play important roles in analysis of DNA chains in Benzer [1], restriction maps of DNA in Waterman and Griggs [11] and other related areas. In this paper, we study a new combinatorial optimization problem, named as the minimum clique partition problem with constrained weight, for interval graphs. For a weighted interval graph G and a bound B, partition the weighted intervals of this graph G into the smallest number of cliques, where each clique, consisting of some intervals whose intersection on a real line is not empty, has its weight not beyond B. We obtain the following results: (1) This problem is N P-hard in the strong sense, and it cannot be approximated within a ratio 3/2 -ε in polynomial-time for any ε > 0; (2) We design some approximation algorithms with different constant ratios to this problem; (3) For the case where all intervals have the same weight, we also design an optimal algorithm to solve the problem in linear time. © Springer-Verlag Berlin Heidelberg 2006.